2.1 Effects of a cumulus ensemble upon the large scale temperature and moisture fields by induced subsidence and detrainment

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1 Atmospheric Sciences 6150 Cloud System Modeling 2.1 Effects of a cumulus ensemble upon the large scale temperature and moisture fields by induced subsidence and detrainment Arakawa (1969, 1972), W. Gray (1971), Ooyama (1971), Pearce and Riehl (1969), and Yanai (1971) were among the first to recognize the importance of cumulus induced subsidence and detrainment of cloud air in modifying the large scale environment. Arakawa (1969) proposed a parameterization scheme for a three-layer general circulation model, in which the role of mass flux in a cloud ensemble and detrainment from the clouds were explicitly included, but only a single cloud type was allowed to occur at a given time and space location. Arakawa (1972) wrote the scheme in a more general continuous form, but the single cloud type assumption was still retained. Yanai (1971) proposed an observational program to obtain the cloud mass flux, combining the large scale heat and moisture budgets with this cloud model. (a) Preliminary considerations For convenience, we define the dry static energy as and the moist static energy as s c p T + gz, (1) h c p T + gz + Lq, (2) where s is the sum of enthalpy and potential energy, and h the sum of s and latent energy. For dry adiabatic processes, ds dt while for both dry and moist adiabatic processes, 0, (3) dh dt 0. (4) We consider an ensemble of cumulus clouds that is embedded in a tropical largescale motion system. The thermodynamic energy equation is ds dt = Q R + L(c e), (5) where Q R is the radiative heating rate, c the rate of condensation, e the rate of evaporation of cloud condensate per unit mass of air. The continuity equation for moisture is dq = e c. (6) dt Page 1

2 With the aid of the (anelastic) continuity equation, h (ρ V h ) + (5) and (6) are written in the flux forms (ρ w) = 0, (7) = ρ Q R + ρ L(c e), (8) where m ρ w is the vertical mass flux per unit area. = ρ (e c), (9) We consider a horizontal area that is large enough to contain an ensemble of cumulus clouds but small enough to be regarded as a fraction of a large scale motion system. Taking horizontal averages of (8) and (9), we obtain = ρ Q R + ρ L( c e ) (s m ), (10) = ρ( e c ) (q m ), (11) where the overbar ( ) is the horizontal average, and ( ) is the deviation from the average. Let σ be the fractional area occupied by active cumulus clouds. (σ = i σ i.) For simplicity, we assume that all clouds have the same characteristic values of s, q, and ρ w at a given level. Then s = σ s c + (1 σ) s, (12) q = σ q c + (1 σ) q, m ρ w = σ m c + (1 σ) m, + M. (13) Cloud System Modeling Page 2

3 M c is called the cloud mass flux, and ( ) is the value in the cloud free environment. s m = σ s c m c + (1 σ) s m, s m = σ 2 s c m c + (1 σ) 2 s m + σ(1 σ)( s m c + s c m). Therefore s m = s m s m = σ(1 σ)(s c s)(m c m). (14) We will make use of the facts: σ << 1 (15a) s c s << s, q c q << q. (15b) [(15b) are strong (sufficient) conditions. They are not necessary.] Then we obtain s = σ(s c s) + s s, q = σ(q c q) + q q, (16) However, m (mean large scale vertical mass flux) m (vertical mass flux of the cloud environment). (Case 1): Therefore and (exact) (Case 2): m = 0 (no net vertical mass flux) m = σ 1 σ m c (compensating downward mass flux) ( m c m = m c 1 + σ ) = m c 1 σ 1 σ. s m = σ m c (s c s) (s c s). m = 0 (all ascent is in clouds only) s m = σ m c (1 σ)(s c s) M c (s c s). (17a) (17b) Therefore in general we have s m σm c (s c s) (s c s). (18) Cloud System Modeling Page 3

4 Therefore (10) and (11) become = ρ Q R + ρ L(c e) [M c(s c s)], (19) apparent heat source = ρ(e c) [M c(q c q)], (20) apparent moisture source where the ( ) symbol has been omitted for Q R, c, and e. From (19) and (20) we have also (ρ h) + h (ρ hv (h m) h ) + = ρ Q R [M c(h c h)], (21) where h = s + Lq (moist static energy). (b) The effects of cumulus induced subsidence and detrainment So far we have expressed the vertical cumulus transports of sensible heat and moisture in terms of the cloud mass flux M c and the excess of s c (thus T c ) and q c from the environmental values. For simplicity let us ignore the radiative heating Q R and evaporation of clouds e. Then (19) and (20) are simplified as = ρ Lc [M c(s c s)], (22) = ρc [M c(q c q)], (23) We assume that the cumulus ensemslbe maintains a heat and moisture balance with the large scale environment. Thus there will be no accumulation of heat or moisture in the cumulus ensemble over a time scale describing the large scale motion. The balance equations for the cumulus ensemble are (ρ σs c ) s (M c s c ) + ρ L c = 0, (heat balance) (24) (ρ σq c ) q (M c q c ) ρ c = 0, (moisture balance) (25) Cloud System Modeling Page 4

5 (ρ σh c ) h (M c h c ) entrainment for the entrainment layer where M c / > 0, and (ρ σs c ) for the detrainment layer where M c / < 0, = 0, (moist static energy balance) (26) s c (M c s c ) + ρ L c = 0, (27) (ρ σq c ) q c (M c q c ) ρ c = 0, (28) (ρ σh c ) h c entrainment (M c h c ) = 0, (29) For the entrainment layer, we substitute (25) into the right hand side of (22) to find ρ L c [ [M Mc c(s c s)] = L q (M c q c ) ] [ ] M c (s c s) L q (M c h c ) + (M c s) h (M c h c ) s +M c (30) }{{ } =0 by (26) Therefore (22) becomes s. (31) Similarly (23) becomes q. (32) For the detrainment layer ( M c / < 0), we obtain from (29) M c h c = 0, (33) ρ L c [ [M Mc c(s c s)] = L q c (M c q c ) ] [ ] M c (s c s) q c = LM c M s c c + M s c (s c s) M c h c s = M c +M c }{{ } (s c s) M c equals 0 (34) Cloud System Modeling Page 5

6 Therefore (22) becomes s (s c s) M c. (35) Similarly (23) becomes q (q c q) M c. (36) For the entrainment layer, (31) shows that the apparent heating (in the absence of radiative heating and evaporative cooling) is due to the adiabatic (compression) warming by the downward motion between the clouds, which is compensating the cumulus mass flux M c (M c s/ > 0). (32) shows that the apparent moisture sink is due to the drying effect of the compensating subsidence (M c q/ < 0). In addition, for the detrainment layer, (35) and (36) show that the detrainment of excess static energy, s c s = c p (T c T ), and excess moisture, q c q, act to warm (if T c T > 0) and moisten (if q c q > 0) the environment. In this simple case, the problem of parameterization of cumulus convection reduces to the determination of M c in terms of the large scale variables. Once M c is known, we can obtain h c from (26). Then using the relations where s c s = c p (T c T ) = γ (h c h ), (37a) q c q = γ L (h c h ), (37b) γ L ( q ) c p T p (Arakawa, 1969), we can determine T c and q c. In the above q ( T, p) is the saturation mixing ratio at ( T, p). The rate of condensation c is determined by (25), because we know M c and q c in addition to q ( q). Cloud System Modeling Page 6